Ritt's theorems on polynomial
decompositions [1]} express the
extent to which a polynomial admits a
unique presentation as a composition of
indecomposable polynomials.
Loosely speaking, if the polynomial
\(P\)  may be expressed in two different
ways as \(P = P_k \circ \cdots \circ P_1\)  and
\(P = Q_\ell \circ \cdots \circ Q_1\)  where each
\(P_i\)  and \(Q_j\)  is indecomposable, then
\(\ell = k\)  and the decomposition
\(\left( Q_k, \ldots , Q_1 \right)\)  may be
obtained from the decomposition
\(\left( P_k, \ldots , P_1 \right)\)  by transforming
these sequences through a finite number of what
in [2]} we call Ritt swaps.
We encode this process of passing from
one decomposition to another via a
sequence of Ritt swaps through a certain
partial monoid action and then take this
further to encode all skew-invariant
curves through more general partial
monoid actions.
